Langevin Algorithms : Questions at the Numerical Analysis / Applied Probability Interface

Randomly evolving phenomena are often mathematically described by so-called Stochastic (Partial) Differential Equations (SPDE), which essentially give update rules for how the system is to evolve in any given infinitesimally small time instance. Mathematicians, statisticians and more generally scientists in many areas are interested in this area as they form plausible models for diverse problems ranging from the price of a stock to the movement of a molecule. They are also of interest in more abstract settings for exploring complex parameter spaces for statistical inference.

Commonly, interest is in how such a system will behave "on average'over a long period of time. This is termed the * steady state' behaviour of the system. For example, a piece of string tied at both ends is subject to constant perturbations from surrounding molecules, and thus continually flutuates, but we might be interested in its average position. If we try to simulate such a random system, we need to follow the update rules in discrete time. This is an approximation of the true continuous-time dynamics, but for sufficiently fine time-discretisation, this approximation ought to have some claim to being a good one. However, the steady state of the system might not be adequately approximated by this method. Fortunately an easy way to correct for the error in estimating averages exists in the guise of the so-called Metropolis-Hastings algorithm which occasionally rejects an update move in favour of staying at the same location.

If we use large time steps in this discretisation, the Metropolis-Hastings rejection moves will be required often, and the overall continuous-time dynamics will not be well approximated. However since interest is in steady state behaviour, this is not necessarily a problem. On the other hand if time steps are too large, then a large proportion of proposed moves will be rejected which will adversely affect the estimation of steady state.

This project is all about devising random algorithms which can efficiently simulate from approximations to SPDEs in order to estimate properties of their steady state behaviour. There are two types of question we will answer. Firstly we will consider the problem of choosing an efficient SPDE which resembles its "average1 behaviour in a relatively short time period. Secondly, we shall consider how to optimally choose the time-dicretisation rules to maximise the efficiency of the algorithm for estimating steady state properties.